Coherence loss in stroboscopic radar ranging in the problem of asteroid size estimation

0.17 MB·

by  V. D. Zakharchenko

Coherence loss in stroboscopic radar ranging
in the problem of asteroid size estimation1
V. D. Zakharchenko2, I. G. Kovalenko3, V. Yu. Ryzhkov4
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VolgogradStateUniversity,UniversitetskijPr.,100,Volgograd400062,Russia
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n Abstract
a
J We consider the problem of coherence loss in a stroboscopic high res-

8 olution radar ranging due to phase instability of the probing and reference
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radiosignals. Requirementstothecoherenceofreferencegeneratorsinstro-

] boscopic signal processing system are formulated. The results of statistical
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modelingarepresented.
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.
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p
Keywords Near-Earth objects, Asteroids, Stroboscopic radar observations, Wide-
-
o bandradiosignals
r
t
s
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[ 1 Preamble

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v
Development of the methods that allow improving accuracy of determining the
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7 asteroidsizes(i.e. whethertheymeasuredozensorhundredsmetersindiameter)is
3 important for correct estimateof damage they can cause (either regional or global
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catastrophes, respectively). At the same time this research can be interesting for
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. specialistswhostudyshapesandthesurfacegeometryofsmallbodiesoftheSolar
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0 system.
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Inourpreviousworks[Zakharchenkoetal.(2015),Zakharchenko&Kovalenko(2014a)]
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: weproposedthemethodtoestimatesizesofpassivecosmicobjectswhichmethod
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i utilizes the radiolocation probing by ultra-high-resolving nanosecond signals for
X
obtaining radar signatures. The method involves radio pulse strobing of reflected
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a ultra-high-resolving signals from the surface of the cosmic object. The complete
coherence of the probing and reflected signals is an essential condition of the
1PreprintsubmittedtoActaAstronautica
2E-mailaddress: zakharchenko [email protected]
3E-mailaddress: [email protected]
4E-mailaddress:[email protected]
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method. However such a condition corresponds to idealized case when no phase
instabilities exist in the signal processing system. The real sources of reference
oscillations have nonzero instant instability of frequency which leads to loss of
coherence at large signal lags (large distances). This factor restricts performance
of coherent processing methods and leads to reduction of signal-to-noise ratio at
theoutputofastroboscopicsystem.
Intheanalysisoftimescaletransformationofbroadbandradiosignals[Zakharchenkoetal.(2015),
Zakharchenko&Kovalenko(2014a)] the complete coherence of carrier frequen-
ciesofthemeasuredandthereferenceoscillationsiscommonlyassumed. Sucha
concept corresponds to an absence of phase instabilities in the signal processing
system. Inrealdevicesthisconditioncanbebrokenduetodeviationsofreference
generator frequency and phase, instabilities of delays in a signal path and other
factors. These factors restrict performance of coherent processing methods and
leadtoreductionofsignal-to-noiseratioattheoutputofastroboscopicsystem.
Let us consider the influence of phase instability of the carrier frequencies of
the measured and the reference signals on statistical characteristics of the trans-
formed signal in stroboscopic processing. We will describe the loss of coherence
by a random process θ(t), i.e. by a fluctuation component of a phase difference
between the received and the strobe radio signals. The statistical characteristics
of the phase difference θ(t) are considered to be known. In the analysis we will
assume that the coefficient of spectral transformation N is large enough to use
asymptoticestimates.
The model of stroboscopic processing of the reflected signals (Fig. 1) differs
from one considered in the work [Zakharchenkoetal.(2015)] by the low-pass fil-
ter being replaced with the tracking filter which adaptively tunes to differential
frequency of carriers Ω=2ω V /c where ω is the carrier frequency of the prob-
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ingsignal,V istheradialvelocityofanasteroid.
r
ForanexactdeterminationofdifferentialfrequencyΩ,theradialvelocityV of
r
an asteroid has to be measured independently using narrow-band methods based
on center of mass of the Doppler signal spectrum. One of such effective methods
is the method of real-time assessment of radial velocity by means of fractional
differentiation of a Doppler signal considered in the previous work of the authors
[Zakharchenko&Kovalenko(2014b)].
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. .
~
x(k,t) y(t) y(t)
Φ
.
a(k,t) Ω
Figure1: ThemodelofstroboscopicprocessingofaDopplersignal.
2 Modeling of coherence loss in stroboscopic signal
processing
Let us represent the complex models of the received x˙(t) and the reference a˙(t)
signals[Zakharchenkoetal.(2015)]includingphaseinstabilityθ(t)intheform
N N
x˙(t)= ∑A(t−kT)ej[ω0t+θ(t)], a˙(t)= ∑A (t−kT )ejω1t, (1)
1 1
k=0 k=0
where A˙(t) and A˙ (t) are the complex envelopes providing high range resolution;
1
T, T are the repetition periods of the signal and the strobe; T =T +∆T; ∆T =
1 1
2TV /cisthesamplingincrement(∆T (cid:28)T,T ).
r 1
The value of stroboscopic sample of a signal in the k-th sampling period can
bepresentedas
1 (cid:90) (k+1)T
y˙ = A(t−kT)A (t−kT )ej[Ωt+θ(t)]dt. (2)
k 1 1
2T
kT
Let us assume θ(t) be a stationary random zero-mean process with correlation
windowτ exceedingthedurationofstrobesignals. Letussupposealsothatslow
θ
phasedisplacementsaretrackedbystabilizingsystem,whereforeonecanneglect
the correlation of adjacent samples {θ } and set (cid:104)θθ (cid:105)=σ2δ . This allows for
k i k θ ik
presentationofthesampley˙ intheform
k
ej[ΩkT1+θk] (cid:90) T
y˙ ≈ A(t(cid:48))A (t(cid:48)−k∆T)dt =y˙ ejθk.
k 1 k0
2T
0
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where θ = θ(kT ) is the sample of the random process θ(t) and y˙ stands for
k 1 k0
the stroboscopic sample (2) when there is no phase instability in the signal pro-
cessingsystem. Toensurethemodeofultra-high-resolutionofradarsignaturesof
asteroids[Zakharchenko&Kovalenko(2014a)]onehastousenanosecondsignals
with pulse ratio of order 103−106, thus, the aforementioned approximations are
perfectlyacceptable.
The average value (mathematical expectation) of samples (2) obtained by av-
eragingoverphaseθ canbeexpressedas
k
My =(cid:104)y˙ (cid:105)=βy˙ , (3)
k k k0
where β =(cid:104)exp[jθ ](cid:105)= χ (1); χ (ν) is the characteristic function of the distri-
k θ θ
bution law of phase fluctuations W(θ ). For the normal process with zero mean
k
W(θ )=N(0,σ2)thisvalueisequaltoβ =exp[−0.5σ2]<1.
k θ θ
Varianceofsamplesy˙ amountstoDy˙ =(cid:104)|y˙ −My˙ |2(cid:105)=|y˙ |2(1−β2)under
k k k k k0
theassumptionsmade. Theratioofvariancetosquaredmeanvalueis
Dy˙ 1−β2
k
η = = (4)
|My˙ |2 β2
k
it has the meaning of relative power level of output noise resulting from phase
fluctuations. This ratio can be significantly reduced by increasing the spectral
transformation coefficient N = T/∆T at the expense of sampling step ∆T de-
crease and by using data storage in a system digital filter. In this case the vari-
ance Dy will be lowered by a factor m where m is the accumulation coefficient
k
[Zakharchenko(1999)]. Given that the filter and the spectrally compressed sig-
nal band are adaptively matched, the value m is asymptotically equivalent to the
numberofsamplingsteps∆T packedinthesignaldurationτ : m∼τ /∆T.
x x
3 Modeling results and quantitative estimates
We performed numerical simulation of processing in the radio pulse strobing
scheme (Fig. 1) of the signal A(t)cos[(ω +Ω)t +θ(t)] reflected by a single
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bright point of the surface of a moving asteroid. The envelopes of the probing
and strobing A(t)cosω t signals were chosen identical: A(t)=A exp[−2(t/τ)2]
0 0
with the effective duration τ determined according to the method of moments
[Gonorovskii(1986)]: τ =2||tA(t)||/||A(t)||. The random process θ(t) was spec-
ified as a sequence of uncorrelated samples θ = θ(kT) with the normal distri-
k
bution: W(θ )=N(0,σ2). The filter’s transfer function was rectangular with the
k θ
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_
E
1.0 y
E
y0
0.9
0.8
0.7
_
σ
0.6 E
E
y0
0.5
m = 2
0.4
0.3 m = 5
0.2
0.1
m = 15
0.0
0 1.0 2.0 3.0 4.0 5.0 6.0 σ , rad
θ
Figure 2: Statistical characteristics of the output signal y (mathematical expecta-
tion and root-mean-square deviation) vs. phase instability σ for different values
θ
ofaccumulationcoefficientm.
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bandwidth equal to the width of the transformed signal power spectrum at 10%
max(20dB)level.
Since the energy of received signal is important for optimal reception under
additive noise conditions [Gonorovskii(1986)], the influence of phase instability
was estimated as a decrease of the mean signals energy E =||y(t)||2 at the out-
y
put of the signal processing filter, relative to the energy E under full coherence
y0
condition(σ =0).
θ
Fig.2demonstratesstatisticalcharacteristicsoftheoutputsignalofthestrobo-
scopicsignalprocessingsystemfordifferentaccumulationcoefficientsinthefilter
obtainedbystatisticalmodelingatspecifiedvaluesΩ=2πF;F =512;τ =0.015;
N =2048. The quantity E /E corresponding to signals energy decrease at the
y y0
output of the band-pass filter of the stroboscopic signal processing system vs.
phase instability is plotted in Fig. 2. The relative errors caused by phase insta-
bility are also shown.. Presented results of statistical modeling are obtained by
averagingover100simulationruns.
Aspreviouslynoted[Zakharchenkoetal.(2015)],foracosmicobjectofabout
50 m in size the range resolution of δr ∼ 0.5 m can be provided by coherent
stroboscopic signal processing of signals with duration ∼ 3 ns in the X-range
(f ∼10GHz)andfrequencyband∆f ∼300MHz.
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As it can be seen from Fig. 2 decrease of mean received signal power by half
correspondstothephaseinstabilityofσ ∼3rad. Thevalueofphasedeviationθ
θ
caused by a short-time frequency instability ∆ω and by finite signal propagation
timet =2R/cis
0
(cid:90) t+t
0
θ(t)≤ ∆ω(t(cid:48))dt(cid:48). (5)
t
It represents the Wiener process [Kazakov(1973)] with normal distribution. The
upper-boundestimateofphasedifferencegives|∆θ|≤max|∆ω(t(cid:48))|t .
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For operation of stroboscopic radar station at range R with acceptable phase
instability of σ <3 rad it is required to ensure |∆ω|<cσ /2R. This condition
θ ∆θ
ensuresthatthenoiseleveldoesnotexceed∼7dBattheaccumulationcoefficient
m>10. Atthecontemporarytechnologylevelthestabilityofreferencegenerators
with relative error no grater than δ ∼10−12, which corresponds to ∼0.01 Hz in
the X-range, is quite realizable [Belov(2004)]. Thus, the system can functionally
operatewithin5millionkmdistance.
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4 Conclusion
Loss of coherence in stroboscopic radar ranging systems caused by phase insta-
bilities of the reference sources leads to sensitivity degradation and is equivalent
totheeffectsofmodulatinginterference. Noisereductionattheoutputofthestro-
boscopic converter caused by loss of coherence can be achieved by reducing the
samplestep∆T =T/N withcorrespondingincreaseoftheprocessingtime.
Acknowledgements
WearegratefultoVitalyKorolevforthehelpatvectorizationofdrawingsand
Victor Levi for careful reading of manuscript. The work is fulfilled within the
frameworkofprojectssupportedbygrantsfromtheRussianFoundationforBasic
Research15-47-02438-r-povolzhie-aand14-02-97001-r-povolzhie-a.
References
[Zakharchenkoetal.(2015)] V.D. Zakharchenko, I.G. Kovalenko, O.V. Pak, Es-
timate of sizes of small asteroids (cosmic bodies) by the method of strobo-
scopicradiolocation,ActaAstronautica108(2015)57-61
[Zakharchenko&Kovalenko(2014a)] V.D. Zakharchenko, I.G. Kovalenko, The
method of counteraction of threat to the Earth by means of estimation of
sizesofpassivecosmicobjects,RFPatentNo2527252,07.06.2013
[Zakharchenko&Kovalenko(2014b)] V.D. Zakharchenko, I.G. Kovalenko, On
protecting the planet against cosmic attack: ultrafast real-time estimate of
theasteroid’sradialvelocity,ActaAstronautica98(2014)158-162
[Zakharchenko(1999)] V.D. Zakharchenko, Self-strobing of fast moving targets
inradioearlywarningsystems,Physicsofwaveprocessesandradiotechnical
systems2(3-4)(1999)34-39(inRussian).
[Gonorovskii(1986)] I.S. Gonorovskii, Radio circuits and signals, Moscow, Ra-
dioisviaz,1986(inRussian).
[Kazakov(1973)] V.A. Kazakov, Introdusction to the theory of Markovian pro-
cesses,Moscow,Sov.Radio,1973(inRussian).
7
[Belov(2004)] L. Belov, Reference generators, Electronicka: Science, Technol-
ogy,Business(6)(2004)38-44(inRussian).
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